Question: Simplify. Rewrite the expression in the form $5^n$. $\left(5^3\right)^{2}=$
Answer: $\begin{aligned} \left(5^3\right)^{2}&=5^{3\cdot 2} \\\\ &=5^{6} \end{aligned}$ This follows from the general rule $\left(x^m\right)^{n}=x^{m\cdot n}$. We can also see this is correct by expanding the powers. $\begin{aligned} \left(5^3\right)^{2}&=\underbrace{5^3\cdot 5^3}_\text{2 times} \\\\\\ &=\underbrace{ \underbrace{5\cdot 5\cdot 5}_\text{3 times} \cdot \underbrace{5\cdot 5\cdot 5}_\text{3 times}} _\text{2 times} \\\\ &=5^{6} \end{aligned}$ In conclusion, $\left(5^3\right)^{2}=5^{6}$.